For the definition of "region" of the set of partitions of j, see A206437.

a(n) is also the number of positive integers in the n-th row of triangle A186114. a(n) is also the number of positive integers in the n-th row of triangle A193870.

Also triangle read by rows: T(j,k) = number of parts in the k-th region of the last section of the set of partitions of j. See example. For more information see A135010.

a(n) is also the length of the n-th vertical line segment in the minimalist diagram of regions and partitions. The length of the n-th horizontal line segment is A141285(n). See also A194447. - Omar E. Pol, Mar 04 2012

In order to construct this sequence with a cellular automaton we use the following rules: We start in the first quadrant of the square grid with no toothpicks. At stage n we place A141285(n) toothpicks of length 1 connected by their endpoints in horizontal direction starting from the point (0, n). Then we place toothpicks of length 1 connected by their endpoints in vertical direction starting from the exposed toothpick endpoint downward up to touch the structure or up to touch the x-axis. a(n) is the number of toothpicks in vertical direction added at n-th stage (see example section and A139250, A225600, A225610).

a(n) is also the length of the n-th descendent line segment in an infinite Dyck path in which the length of the n-th ascendent line segment is A141285(n). See Example section. For more information see A211978, A220517, A225600.

The next figure shows a minimalist diagram of the first seven regions. The n-th horizontal line segment has length A141285(n). a(n) is the length of the n-th vertical line segment, which is the vertical line segment ending in row n (see also A225610).

. _ _ _ _ _

. 7 _ _ _ |

. 6 _ _ _|_ |

. 5 _ _ | |

. 4 _ _|_ | |

. 3 _ _ | | |

. 2 _ | | | |

. 1 | | | | |

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. 1 2 3 4 5

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Illustration of initial terms from an infinite Dyck path in which the length of the n-th ascendent line segment is A141285(n). a(n) is the length of the n-th descendent line segment.